Generalizations of Cyclostationary Signal Processing: Spectral Analysis and Applications -- Contents -- About the Author -- Acknowledgements -- Preface -- List of Abbreviations -- 1 Background -- 1.1 Second-Order Characterization of Stochastic Processes -- 1.1.1 Time-Domain Characterization -- 1.1.2 Spectral-Domain Characterization -- 1.1.3 Time-Frequency Characterization -- 1.1.4 Wide-Sense Stationary Processes -- 1.1.5 Evolutionary Spectral Analysis -- 1.1.6 Discrete-Time Processes -- 1.1.7 Linear Time-Variant Transformations -- 1.2 Almost-Periodic Functions -- 1.2.1 Uniformly Almost-Periodic Functions -- 1.2.2 AP Functions in the Sense of Stepanov, Weyl, and Besicovitch -- 1.2.3 Weakly AP Functions in the Sense of Eberlein -- 1.2.4 Pseudo AP Functions -- 1.2.5 AP Functions in the Sense of Hartman and Ryll-Nardzewski -- 1.2.6 AP Functions Defined on Groups and with Values in Banach and Hilbert Spaces -- 1.2.7 AP Functions in Probability -- 1.2.8 AP Sequences -- 1.2.9 AP Sequences in Probability -- 1.3 Almost-Cyclostationary Processes -- 1.3.1 Second-Order Wide-Sense Statistical Characterization -- 1.3.2 Jointly ACS Signals -- 1.3.3 LAPTV Systems -- 1.3.4 Products of ACS Signals -- 1.3.5 Cyclic Statistics of Communications Signals -- 1.3.6 Higher-Order Statistics -- 1.3.7 Cyclic Statistic Estimators -- 1.3.8 Discrete-Time ACS Signals -- 1.3.9 Sampling of ACS Signals -- 1.3.10 Multirate Processing of Discrete-Time ACS Signals -- 1.3.11 Applications -- 1.4 Some Properties of Cumulants -- 1.4.1 Cumulants and Statistical Independence -- 1.4.2 Cumulants of Complex Random Variables and Joint Complex Normality -- 2 Generalized Almost-Cyclostationary Processes -- 2.1 Introduction -- 2.2 Characterization of GACS Stochastic Processes -- 2.2.1 Strict-Sense Statistical Characterization -- 2.2.2 Second-Order Wide-Sense Statistical Characterization.

2.2.3 Second-Order Spectral Characterization -- 2.2.4 Higher-Order Statistics -- 2.2.5 Processes with Almost-Periodic Covariance -- 2.2.6 Motivations and Examples -- 2.3 Linear Time-Variant Filtering of GACS Processes -- 2.4 Estimation of the Cyclic Cross-Correlation Function -- 2.4.1 The Cyclic Cross-Correlogram -- 2.4.2 Mean-Square Consistency of the Cyclic Cross-Correlogram -- 2.4.3 Asymptotic Normality of the Cyclic Cross-Correlogram -- 2.5 Sampling of GACS Processes -- 2.6 Discrete-Time Estimator of the Cyclic Cross-Correlation Function -- 2.6.1 Discrete-Time Cyclic Cross-Correlogram -- 2.6.2 Asymptotic Results as N → ∞ -- 2.6.3 Asymptotic Results as N → ∞ and Ts → 0 -- 2.6.4 Concluding Remarks -- 2.7 Numerical Results -- 2.7.1 Aliasing in Cycle-Frequency Domain -- 2.7.2 Simulation Setup -- 2.7.3 Cyclic Correlogram Analysis with Varying N -- 2.7.4 Cyclic Correlogram Analysis with Varying N and Ts -- 2.7.5 Discussion -- 2.7.6 Conjecturing the Nonstationarity Type of the Continuous-Time Signal -- 2.7.7 LTI Filtering of GACS Signals -- 2.8 Summary -- 3 Complements and Proofs on Generalized Almost-Cyclostationary Processes -- 3.1 Proofs for Section 2.2.2 "Second-Order Wide-Sense Statistical Characterization" -- 3.1.1 Proof of Theorem 2.2.17 Mean-Square Integrability of GACS Processes -- 3.1.2 Proof of Fact 2.2.18 -- 3.1.3 Proof of Fact 2.2.19 -- 3.1.4 Proof of Fact 2.2.20 -- 3.2 Proofs for Section 2.2.3 "Second-Order Spectral Characterization" -- 3.2.1 The μ Functional -- 3.2.2 Proof of Theorem 2.2.22 Loève Bifrequency Spectrum of GACS Processes -- 3.3 Proofs for Section 2.3 "Linear Time-Variant Filtering of GACS Processes" -- 3.3.1 Proof of (2.112) -- 3.3.2 Proof of (2.117) -- 3.4 Proofs for Section 2.4.1 "The Cyclic Cross-Correlogram" -- 3.4.1 Proof of Theorem 2.4.6 Expected Value of the Cyclic Cross-Correlogram.

3.4.2 Proof of Theorem 2.4.7 Covariance of the Cyclic Cross-Correlogram -- 3.5 Proofs for Section 2.4.2 "Mean-Square Consistency of the Cyclic Cross-Correlogram" -- 3.5.1 Proof of Theorem 2.4.11 Asymptotic Expected Value of the Cyclic Cross-Correlogram -- 3.5.2 Proof of Theorem 2.4.12 Rate of Convergence of the Bias of the Cyclic Cross-Correlogram -- 3.5.3 Proof of Theorem 2.4.13 Asymptotic Covariance of the Cyclic Cross-Correlogram -- 3.5.4 Proof of Corollary 2.4.14 -- 3.6 Proofs for Section 2.4.3 "Asymptotic Normality of the Cyclic Cross-Correlogram" -- 3.6.1 Proof of Lemma 2.4.17 -- 3.6.2 Proof of Theorem 2.4.18 Asymptotic Joint Normality of the Cyclic Cross-Correlograms -- 3.7 Conjugate Covariance -- 3.8 Proofs for Section 2.5 "Sampling of GACS Processes" -- 3.8.1 Proof of Theorem 2.5.3 Aliasing Formula for the Cyclic Cross-Correlation Function -- 3.9 Proofs for Section 2.6.1 "Discrete-Time Cyclic Cross-Correlogram" -- 3.9.1 Proof of Theorem 2.6.2 Expected Value of the Discrete-Time Cyclic Cross-Correlogram -- 3.9.2 Proof of Theorem 2.6.3 Covariance of the Discrete-Time Cyclic Cross-Correlogram -- 3.10 Proofs for Section 2.6.2 "Asymptotic Results as N → ∞ " -- 3.10.1 Proof of Theorem 2.6.4 Asymptotic Expected Value of the Discrete-Time Cyclic Cross-Correlogram -- 3.10.2 Proof of Theorem 2.6.5 Asymptotic Covariance of the Discrete-Time Cyclic Cross-Correlogram -- 3.10.3 Proof of Theorem 2.6.7 Rate of Convergence of the Bias of the Discrete-Time Cyclic Cross-Correlogram -- 3.10.4 Proof of Lemma 2.6.9 Rate of Convergence to Zero of Cumulants of Discrete-Time Cyclic Cross-Correlograms -- 3.10.5 Proof of Theorem 2.6.10 Asymptotic Joint Normality of the Discrete-Time Cyclic Cross-Correlograms -- 3.11 Proofs for Section 2.6.3 "Asymptotic Results as N → ∞ and Ts → 0" -- 3.11.1 Proof of Lemma 2.6.12.

3.11.2 Proof of Theorem 2.6.13 Mean-Square Consistency of the Discrete-Time Cyclic Cross-Correlogram -- 3.11.3 Noninteger Time-Shift -- 3.11.4 Proof of Theorem 2.6.16 Asymptotic Expected Value of the Hybrid Cyclic Cross-Correlogram -- 3.11.5 Proof of Theorem 2.6.17 Rate of Convergence of the Bias of the Hybrid Cyclic Cross-Correlogram -- 3.11.6 Proof of Theorem 2.6.18 Asymptotic Covariance of the Hybrid Cyclic Cross-Correlogram -- 3.11.7 Proof of Lemma 2.6.19 Rate of Convergence to Zero of Cumulants of Hybrid Cyclic Cross-Correlograms -- 3.11.8 Proof of Theorem 2.6.20 Asymptotic Joint Normality of the Hybrid Cyclic Cross-Correlograms -- 3.12 Proofs for Section 2.6.4 "Concluding Remarks" -- 3.12.1 Proof of Theorem 2.6.21 Asymptotic Discrete-Time Cyclic Cross-Correlogram -- 3.13 Discrete-Time and Hybrid Conjugate Covariance -- 4 Spectrally Correlated Processes -- 4.1 Introduction -- 4.2 Characterization of SC Stochastic Processes -- 4.2.1 Second-Order Characterization -- 4.2.2 Relationship among ACS, GACS, and SC Processes -- 4.2.3 Higher-Order Statistics -- 4.2.4 Motivating Examples -- 4.3 Linear Time-Variant Filtering of SC Processes -- 4.3.1 FOT-Deterministic Linear Systems -- 4.3.2 SC Signals and FOT-Deterministic Systems -- 4.4 The Bifrequency Cross-Periodogram -- 4.5 Measurement of Spectral Correlation - Unknown Support Curves -- 4.6 The Frequency-Smoothed Cross-Periodogram -- 4.7 Measurement of Spectral Correlation - Known Support Curves -- 4.7.1 Mean-Square Consistency of the Frequency-Smoothed Cross-Periodogram -- 4.7.2 Asymptotic Normality of the Frequency-Smoothed Cross-Periodogram -- 4.7.3 Final Remarks -- 4.8 Discrete-Time SC Processes -- 4.9 Sampling of SC Processes -- 4.9.1 Band-Limitedness Property -- 4.9.2 Sampling Theorems -- 4.9.3 Illustrative Examples -- 4.10 Multirate Processing of Discrete-Time Jointly SC Processes.

4.10.1 Expansion -- 4.10.2 Sampling -- 4.10.3 Decimation -- 4.10.4 Expansion and Decimation -- 4.10.5 Strictly Band-Limited SC Processes -- 4.10.6 Interpolation Filters -- 4.10.7 Decimation Filters -- 4.10.8 Fractional Sampling Rate Converters -- 4.11 Discrete-Time Estimators of the Spectral Cross-Correlation Density -- 4.12 Numerical Results -- 4.12.1 Simulation Setup -- 4.12.2 Unknown Support Curves -- 4.12.3 Known Support Curves -- 4.13 Spectral Analysis with Nonuniform Frequency Spacing -- 4.14 Summary -- 5 Complements and Proofs on Spectrally Correlated Processes -- 5.1 Proofs for Section 4.2 "Characterization of SC Stochastic Processes" -- 5.1.1 Proof of Theorem 4.2.9 Second-Order Temporal Cross-Moment of Jointly SC Processes -- 5.1.2 Proof of Theorem 4.2.10 Time-Variant Cross-Spectrum of Jointly SC Processes -- 5.1.3 Proof of (4.92) -- 5.2 Proofs for Section 4.4 "The Bifrequency Cross-Periodogram" -- 5.2.1 Proof of Lemma 4.4.6 Expected Value of the Bifrequency Cross-Periodogram -- 5.2.2 Proof of Lemma 4.4.7 Covariance of the Bifrequency Cross-Periodogram -- 5.2.3 Proof of Theorem 4.4.8 Asymptotic Expected Value of the Bifrequency Cross-Periodogram -- 5.2.4 Proof of Theorem 4.4.9 Asymptotic Covariance of the Bifrequency Cross-Periodogram -- 5.3 Proofs for Section 4.5 "Measurement of Spectral Correlation - Unknown Support Curves" -- 5.3.1 Proof of Lemma 4.5.5 Expected Value of the Time-Smoothed Bifrequency Cross-Periodogram -- 5.3.2 Proof of Lemma 4.5.6 Covariance of the Time-Smoothed Bifrequency Cross-Periodogram -- 5.3.3 Proof of Theorem 4.5.7 Asymptotic Expected Value of the Time-Smoothed Bifrequency Cross-Periodogram -- 5.3.4 Proof of Corollary 4.5.8 -- 5.3.5 Proof of Theorem 4.5.9 Asymptotic Covariance of the Time-Smoothed Bifrequency Cross-Periodogram -- 5.4 Proofs for Section 4.6 "The Frequency-Smoothed Cross-Periodogram".

5.4.1 Proof of Theorem 4.6.3 Expected Value of the Frequency-Smoothed Cross-Periodogram.